Math/Stats 525: Probability Theory

Winter 2007

Course meets:
Tuesday and Thursday 4:10-5:30, 1060 East Hall.

Instructor: Sergey Fomin, 2858 East Hall, 764-6297, fomin@umich.edu

Office hours: Tuesday 5:40-7:00 and Thursday 11:40-1:00 in 2858 East Hall.

Grader: Paul Siegel, siegelp@umich.edu.

Course homepage: http://www.math.lsa.umich.edu/~fomin/525w07.html

Text (required): Geoffrey Grimmett and David Stirzaker, Probability and Random Processes, 3rd edition, Oxford University Press, 2001.

Where can I buy this textbook?

Supplementary texts (not required):
Geoffrey Grimmett and David Stirzaker, One Thousand Exercises in Probability, 2nd edition, Oxford University Press, 2001.
Sheldon Ross, Introduction to Probability Models, 8th/9th edition, Academic Press, 2002/2006.
Sheldon Ross, A First Course in Probability, 6th/7th edition, Prentice-Hall, 2002/2006.

Prerequisites: Math 450 or Math 451 (preferred) and some exposure to elementary probability and combinatorics.

From departmental course description:
This introductory course in probability theory is more theoretical than Math 425, and requires a stronger mathematical background. No measure theory is assumed. Topics include: probability spaces, discrete and continuous random variables, joint distributions and conditional expectations, characteristic functions, central limit theorem, random walk.

Grade will be based on two 1.5-hour midterm exams, 30% each; 40% homework and quizzes.
Your lowest homework/quiz score will be dropped.

This course will not be graded on a curve, i.e., there are not a set number of each grade to be given out. Every student with the total score of 90% (resp., 80%, 70%, 60%) is guaranteed the final grade of A (resp., B or higher, C or higher, D or higher).

Exams

The midterm exams are held in class. No makeups will be given.
Exams are closed book, closed notebook.
You will be allowed to bring a 3-by-5 index card to the 1st midterm, and two such cards to the 2nd midterm.
First midterm: March 6.
Second midterm: April 17.
Exam preparation: see below for practice problems and other potentially useful links

Textbooks freely available online:
C.M.Grinstead and J.L.Snell, Introduction to Probability.
J.Walrand, Probability and Random Processes .

Other online resources:
Virtual Laboratories in Probability and Statistics
List of web resources in statistics and probability
Basic counting techniques
Proof of Stirling's formula

Practice problems (with dates when they were suggested in class):

January 11: 1.2.3, 1.3.3, 1.3.4, 1.4.3, 1.4.7, 1.8.12, 1.8.22, 1.8.24, 1.8.28
January 18: 1.5.7, 1.5.9, 1.7.1, 1.7.3, 1.7.5, 1.8.20, 1.8.21, 1.8.35, 1.8.39
January 25: 2.7.15, 3.1.1(a,b,d), 3.1.3, 3.2.1, 3.2.2(a,b,c), 3.2.4(a)
February 1: 3.3.2, 3.5.1, 3.11.7, 3.11.11, 3.11.13(b), 3.11.21(a), 3.11.24
February 8: 3.3.3, 3.4.3, 3.4.8, 3.6.8, 3.11.8, 3.11.12, 3.11.16
February 15: 3.6.3, 3.7.4, 3.7.5(b), 3.11.4, 3.11.6(b)
March 8, 15: 4.1.1(a), 4.1.2, 4.2.2, 4.3.3 (r=1), 4.13.14, 5.8.9(a), 5.10.1(a), 5.12.25(a), 5.12.33(a)
March 22: 4.4.3, 4.4.5, 4.7.3, 4.7.4(a), 4.7.13 (1st part), 5.10.2, 5.12.5 (1st part)
March 29: 4.5.4, 4.7.7(c), 4.7.11, 4.8.3, 4.14.7, 4.14.54, 4.14.55 (1st part)
April 5: 4.6.1, 4.6.4, 4.6.9, 4.7.8, 4.7.10

Preparing for the First Exam:
The exam will cover Chapters 1-3 in Grimmett and Stirzaker (discrete distributions only). Besides the practice problems listed above, I recommend problems 1-24, 42, 46, 47, 50, 51, 54 in this set of exercises. The answers can be found here. This web page has some complete solutions, and a couple of old exams. (Skip the problems involving continuous distributions.)

Preparing for the Second Exam:
The exam will roughly cover Sections 4.1-4.8, 5.7-5.10 in Grimmett and Stirzaker. Besides the practice problems listed above, I recommend problems 29-39, 41, 43-45, 48-49, 52-53 in this set of exercises. The answers can be found here. This web page has some complete solutions, and a couple of old exams.